Learning outcomes

  • Understand key timeseries concepts and features
  • See example timeseries that we'll use on the course
  • Understand the types of analysis that we want to do, and why

What is a time series?

  • A collection of data indexed by (strictly increasing) time
  • Can be regular or irregular
  • Usually one value per time

A short history of time series

“With one dimension marching along to the regular rhythm of seconds, minutes, hours, days, weeks, months, years, centuries, or millennia, the natural ordering of the time scale gives this design a strength and efficiency of interpretation found in no other graphic arrangement.”

Edward R. Tufte The Visual Display of Quantitative Information p. 28

The first time series example?

Skipping forward a bit …

Diagram showing the distance of the planets to the earth in 1732, also showing a complete lunar eclipse and a partial solar eclipse in that year Nicolaas Kruik 1678 - 1754 Dutch Astronomer & Meteorologist

Lambert

A graph of solar warming vs. lattitude.
Johann Heinrich Lambert 1728 - 1777

Application to economic data

Willaim Playfair's trade-balance time-series chart, published in his Commercial and Political Atlas, 1786

Time Series Analysis: What's the point?

What do we use time series methods for? Often, we are trying to do at least one of the following:

  1. Description: what is going on?
  2. Understanding: how is it going on, and why?
  3. Prediction: What is going to go on next?

First, description

  • HadCRUT4 Annual globally averaged surface temperature, from the Met Office Hadley Centre and Climate Research Unit, UEA.

  • How do we separate the structure (here the long term trend) and the random noise?

Plot the data

  • Using a line plot gives a slightly different impression.

What is going on?

We get a different impression again if we just plot the last 15 years ..
(GWPF + hiatus)

And make a prediction

Would we/could we have predicted the hiatus? How about the massive jump in temperature of the last year?

So, what is structure? And what is random?

A simpler example? Sea level rise

  • Church and White
  • There appears relatively less noise in this data compared to the trend

Changes in variability & Heteroskedasticity

  • This AirPassengers data set shows heteroskedasticity

Example - GISP2 stable isotope ratios in Greenland ice

  • This data shows heteroskedacity and regime changes

Changes in variability & Heteroskedascticity

Monthly NH sea ice Anomaly from 1978 to present.

Seasonal (and longer) cycles, periodicity

plot(ldeaths)

#with(ldeaths,plot(Age,d18o,type='l'))

HADCET

Monthly Central England Temperature (CET) from 1659

  • Seasonality [on short term] & trend?

Sea ice from 1953

  • Trend, seasonality, heteroskedacity?, break points?

Unpredictability and stochasticity

D18O? Global financial crash? Regime changes

Autocorrelation and memory

  • Autocorrelation is the correlation of a random process with itself at a different time.
  • some created examples here

Stationarity

  • In a stationary timeseries mean, variance and autocorrelation structure do not change over time.

Differencing

  • A timeseries of the differences between the values.
  • Differencing can be used used to make a non-stationary trend stationary.

Other important concepts

  • Stationarity
  • Lag
  • Detrending and differencing
  • Seasonality removal
  • Smoothing & filtering
  • Stochastic processes & Random walks

What kind of analysis might we want to do?

Understanding an underlying system

Prediction of the system

The limits of our analysis

This is the lynx data set, annual numbers of lynx trappings for Canada 1821 - 1934. Regarded as representative of the population.

plot(lynx)

The limits of our analysis